The non-vanishing mass modifies the dispersion relation of the gravitational waves, and the group velocity is constrained with high precision \(-3\cdot 10^{-15}<c_{\text{gw}}-1<7\cdot 10^{-16}\)  at low redshift regime by the multi-messenger observation GW170817 [1]}. It gives us an upper bound on the graviton mass around \(m_g<10^{-23}\text{eV}\) . Our graviton mass is much lower than this upper bound by around 10 orders of magnitude. It remains challenging to directly probe the non-vanishing graviton mass effects at the late time epoch. However, the size of the future event horizon is much shorter during the early universe, and thus it leads to a much larger graviton mass, which may give rise to some interesting observational effects on the stochastic gravitational waves background.
